Related papers: Elementary symmetric polynomials in Stanley--Reisn…

Stanley introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. Stanley later gives a conjectured combinatorial interpretation for the coefficients of the…

Linear Maxwell equations for transverse magnetic (TM) polarized fields support single frequency surface plasmon polaritons (SPPs) localized at the interface of a metal and a dielectric. Metals are typically dispersive, i.e. the dielectric…

Motivated by Stanley's generalization of the chromatic polynomial of a graph to the chromatic symmetric function, we introduce the characteristic polynomial of a representation of the symmetric group, or more generally, of a symmetric…

For any unbranched double covering of compact Riemann surfaces, we study the associated character varieties that are unitary in the global sense, which we call $\text{GL}_n\rtimes\!<\!\sigma\!>\!~$-character varieties. We introduce $k>0$…

We construct type A partially-symmetric Macdonald polynomials $P_{(\lambda \mid \gamma)}$, where $\lambda \in \mathbb{Z}_{\geq 0}^{n-k}$ is a partition and $\gamma \in \mathbb{Z}_{\geq 0}^k$ is a composition. These are polynomials which are…

It is proved that a certain symmetric sequence of nonnegative integers arising in the enumeration of magic squares of given size n by row sums or, equivalently, in the generating function of the Ehrhart polynomial of the polytope of doubly…

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over the field $\mathbb{K}$. In this paper, it is shown that Stanley's conjecture holds for $S/I$, if $I$ is a weakly polymatroidal ideal.

In the intersection of the theories of nonsymmetric Jack polynomials in $N$ variables and representations of the symmetric groups $\mathcal{S}_{N}$ one finds the singular polynomials. For certain values of the parameter $\kappa$ there are…

For each sequence of polynomials, P=(p_1(t),p_2(t),...), we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S^3, such a sequence of polynomials arises naturally as the orders of certain…

We show that the cohomology algebra of the complement of a coordinate subspace arrangement in m-dimensional complex space is isomorphic to the cohomology algebra of Stanley-Reisner face ring of a certain simplicial complex on m vertices.…

We study the geometry of the image of the nonnegative orthant under the power-sum map and the elementary symmetric polynomials map. After analyzing the image in finitely many variables, we concentrate on the limit as the number of variables…

It is shown that two vectors with coordinates in the finite $q$-element field of characteristic $p$ belong to the same orbit under the natural action of the symmetric group if each of the elementary symmetric polynomials of degree…

We consider a class of generally non-self-adjoint eigenvalue problems which are nonlinear in the solution as well as in the eigenvalue parameter ("doubly" nonlinear). We prove a bifurcation result from simple isolated eigenvalues of the…

We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement,…

There is a simple formula for the Ehrhart polynomial of a cyclic polytope. The purpose of this paper is to show that the same formula holds for a more general class of polytopes, lattice-face polytopes. We develop a way of decomposing any…

Let $P$ be a convex $d$-polytope and $0 \leq k \leq d-1$. In 2023, this author proved the following inequalities, resolving a question of B\'ar\'any: \[ \frac{f_k(P)}{f_0(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose k} +…

One of basic difficulties of machine learning is handling unknown rotations of objects, for example in image recognition. A related problem is evaluation of similarity of shapes, for example of two chemical molecules, for which direct…

Let $\mathcal{P}$ be the class of combinatorial 3-dimensional simple polytopes $P$, different from a tetrahedron, without 3- and 4-belts of facets. By the results of Pogorelov and Andreev, a polytope $P$ admits a realisation in Lobachevsky…

Quantum knot invariants (like colored HOMFLY-PT or Kauffman polynomials) are a distinguished class of non-perturbative topological invariants. Any known way to construct them (via Chern-Simons theory or quantum R-matrix) starts with a…

We study the extent of independence needed to approximate the product of bounded random variables in expectation, a natural question that has applications in pseudorandomness and min-wise independent hashing. For random variables whose…